Find the minimum value

**geton** · June 16th 2008, 02:55 AM

Find the minimum value of (5 cosh x + 3 sinh x).

How can I solve this?

**mr fantastic** · June 16th 2008, 03:08 AM

Originally Posted by geton

Find the minimum value of (5 cosh x + 3 sinh x).

How can I solve this?

Differentiate, put to zero, test nature .... the usual deal.

Note that to solve when the derivative is equal to zero, you should:

1. Replace cosh and sinh with their exponential definitions.
2. Multiply through by 2 e^{-x}.
3. Solve the resulting quadratic equation in e^x .....

**kalagota** · June 16th 2008, 03:14 AM

$\cosh x = \frac{e^x + e^{-x}}{2}$
$\sinh x = \frac{e^x - e^{-x}}{2}$

so, $f(x) = 5\cosh x + 3 \sinh x$

$f'(x) = 5\sinh x + 3\cosh x = \frac{8e^x - 2e^{-x}}{2} = 4e^x - e^{-x} = \frac{4e^{2x} - 1}{e^x}$

we want to find $x$ such that $f'(x) = 0$
since $e^x$ is never zero, we will solve $4e^{2x} - 1 = 0$ .. solving, we shall obtain

$x = -\ln 2$ ... substitute that $x$ to find the minimum value..

--------------------------
to check, $f''(x) = f(x) = 5\cosh x + 3 \sinh x = 4e^x + e^{-x} > 0$ for any $x$ .
thus, $x = -\ln 2$ gives the minimum value for $f(x)$

**geton** · June 16th 2008, 03:30 AM

Yes I got x = - ln2.

But its answer is 4.

I've no idea how can I get this.

**Moo** · June 16th 2008, 03:35 AM

Originally Posted by geton

Yes I got x = - ln2.

But its answer is 4.

I've no idea how can I get this.

Yes, the correct answer is 4.

$f(-\ln 2)=4$

-ln(2) is not the minimum value of 5cosh(x)+3sinh(x), it's just x such that the function is at its minimum.

Got the difference ?

**geton** · June 16th 2008, 03:55 AM

Originally Posted by Moo

Yes, the correct answer is 4.

$f(-\ln 2)=4$

-ln(2) is not the minimum value of 5cosh(x)+3sinh(x), it's just x such that the function is at its minimum.

Got the difference ?

Yup

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